1. Rotational Motion and The Law of Gravity
Chapter Seven
Rotational Motion and
The Law of Gravity

2. Rotational Motion An important part of everyday life Angular motion
Motion of the Earth
Rotating wheels
Angular motion
Expressed in terms of
Angular speed
Angular acceleration
Centripetal acceleration
Introduction

3. Gravity
Rotational motion combined with Newton’s Law of Universal Gravity and Newton’s Laws of motion can explain aspects of space travel and satellite motion
Kepler’s Three Laws of Planetary Motion
Formed the foundation of Newton’s approach to gravity
Introduction

4. Angular Motion Will be described in terms of
Angular displacement, Δθ
Angular velocity, ω
Angular acceleration, α
Analogous to the main concepts in linear motion
Section 7.1

5. The Radian The radian is a unit of angular measure
The radian can be defined as the arc length s along a circle divided by the radius r
Section 7.1

6. More About Radians Comparing degrees and radians
Converting from degrees to radians
Section 7.1

7. Angular Displacement Axis of rotation is the center of the disk
Need a fixed reference line
During time t, the reference line moves through angle θ
The angle, θ, measured in radians, is the angular position
Section 7.1

8. Rigid Body
Every point on the object undergoes circular motion about the point O
All parts of the object of the body rotate through the same angle during the same time
The object is considered to be a rigid body
This means that each part of the body is fixed in position relative to all other parts of the body
Section 7.1

9. Angular Displacement, cont.
The angular displacement is defined as the angle the object rotates through during some time interval
The unit of angular displacement is the radian
Each point on the object undergoes the same angular displacement
Section 7.1

10. Average Angular Speed
The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval
Section 7.1

11. Angular Speed, cont.
The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero
SI unit: radians/sec
rad/s
Speed will be positive if θ is increasing (counterclockwise)
Speed will be negative if θ is decreasing (clockwise)
When the angular speed is constant, the instantaneous angular speed is equal to the average angular speed
Section 7.1

12. Average Angular Acceleration
An object’s average angular acceleration αav during time interval Δt is the change in its angular speed Δω divided by Δt:
Section 7.1

13. Angular Acceleration, cont
SI unit: rad/s²
Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction
When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration
The tangential (linear) speed and acceleration will depend on the distance from a given point to the axis of rotation
Section 7.1

14. Angular Acceleration, final
The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero
Section 7.1

15. Analogies Between Linear and Rotational Motion
There are many parallels between the motion equations for rotational motion and those for linear motion
Every term in a given linear equation has a corresponding term in the analogous rotational equations
Section 7.2

16. Relationship Between Angular and Linear Quantities
Displacements
s = θ r
Speeds
vt = ω r
Accelerations
at = α r
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion
Section 7.3

17. Centripetal Acceleration
An object traveling in a circle, even though it moves with a constant speed, will have an acceleration
The centripetal acceleration is due to the change in the direction of the velocity
Section 7.4

18. Centripetal Acceleration, cont.
Centripetal refers to “center-seeking”
The direction of the velocity changes
The acceleration is directed toward the center of the circle of motion
Section 7.4

19. Centripetal Acceleration, final
The magnitude of the centripetal acceleration is given by
This direction is toward the center of the circle
Section 7.4

20. Centripetal Acceleration and Angular Velocity
The angular velocity and the linear velocity are related (v = r ω)
The centripetal acceleration can also be related to the angular velocity
Section 7.4

21. Total Acceleration
The tangential component of the acceleration is due to changing speed
The centripetal component of the acceleration is due to changing direction
Total acceleration can be found from these components
Section 7.4

22. Vector Nature of Angular Quantities
Angular displacement, velocity and acceleration are all vector quantities
Direction can be more completely defined by using the right hand rule
Grasp the axis of rotation with your right hand
Wrap your fingers in the direction of rotation
Your thumb points in the direction of ω
Section 7.4

23. Velocity Directions, Example
In a, the disk rotates counterclockwise, the direction of the angular velocity is out of the page
In b, the disk rotates clockwise, the direction of the angular velocity is into the page
Section 7.4

24. Acceleration Directions
If the angular acceleration and the angular velocity are in the same direction, the angular speed will increase with time
If the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time
Section 7.4

25. Forces Causing Centripetal Acceleration
Newton’s Second Law says that the centripetal acceleration is accompanied by a force
FC = maC
FC stands for any force that keeps an object following a circular path
Tension in a string
Gravity
Force of friction
Section 7.4

26. Centripetal Force Example
A puck of mass m is attached to a string
Its weight is supported by a frictionless table
The tension in the string causes the puck to move in a circle
Section 7.4

27. Centripetal Force General equation
If the force vanishes, the object will move in a straight line tangent to the circle of motion
Centripetal force is a classification that includes forces acting toward a central point
It is not a force in itself
A centripetal force must be supplied by some actual, physical force
Section 7.4

28. Problem Solving Strategy
Draw a free body diagram, showing and labeling all the forces acting on the object(s)
Choose a coordinate system that has one axis perpendicular to the circular path and the other axis tangent to the circular path
The normal to the plane of motion is also often needed
Section 7.4

29. Problem Solving Strategy, cont.
Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, FC)
The net radial force causes the centripetal acceleration
Use Newton’s second law
The directions will be radial, normal, and tangential
The acceleration in the radial direction will be the centripetal acceleration
Solve for the unknown(s)
Section 7.4

30. Applications of Forces Causing Centripetal Acceleration
Many specific situations will use forces that cause centripetal acceleration
Level curves
Banked curves
Horizontal circles
Vertical circles
Section 7.4

31. Level Curves
Friction is the force that produces the centripetal acceleration
Can find the frictional force, µ, or v
Section 7.4

32. Banked Curves
A component of the normal force adds to the frictional force to allow higher speeds
Section 7.4

33. Vertical Circle Look at the forces at the top of the circle
The minimum speed at the top of the circle can be found
Section 7.4

34. Forces in Accelerating Reference Frames
Distinguish real forces from fictitious forces
“Centrifugal” force is a fictitious force
It most often is the absence of an adequate centripetal force
Arises from measuring phenomena in a noninertial reference frame
Section 7.4

35. Newton’s Law of Universal Gravitation
If two particles with masses m1 and m2 are separated by a distance r, then a gravitational force acts along a line joining them, with magnitude given by
Section 7.5

36. Universal Gravitation, 2
G is the constant of universal gravitational
G = x N m² /kg²
This is an example of an inverse square law
The gravitational force is always attractive
Section 7.5

37. Universal Gravitation, 3
The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1
The forces form a Newton’s third law action-reaction
Section 7.5

38. Universal Gravitation, 4
The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center
This is called Gauss’ Law
Section 7.5

39. Gravitation Constant Determined experimentally Henry Cavendish
1798
The light beam and mirror serve to amplify the motion
Section 7.5

40. Applications of Universal Gravitation
Acceleration due to gravity
g will vary with altitude
In general,
Section 7.5

41. Gravitational Potential Energy
PE = mgh is valid only near the earth’s surface
For objects high above the earth’s surface, an alternate expression is needed
With r > Rearth
Zero reference level is infinitely far from the earth
Section 7.5

42. Escape Speed
The escape speed is the speed needed for an object to soar off into space and not return
For the earth, vesc is about 11.2 km/s
Note, v is independent of the mass of the object
Section 7.5

43. Various Escape Speeds
The escape speeds for various members of the solar system
Escape speed is one factor that determines a planet’s atmosphere
Section 7.5

44. Kepler’s Laws
All planets move in elliptical orbits with the Sun at one of the focal points.
A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals.
The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.
Section 7.6

45. Kepler’s Laws, cont. Based on observations made by Brahe
Newton later demonstrated that these laws were consequences of the gravitational force between any two objects together with Newton’s laws of motion
Section 7.6

46. Kepler’s First Law
All planets move in elliptical orbits with the Sun at one focus.
Any object bound to another by an inverse square law will move in an elliptical path
Second focus is empty
Section 7.6

47. Kepler’s Second Law
A line drawn from the Sun to any planet will sweep out equal areas in equal times
Area from A to B and C to D are the same
The planet moves more slowly when farther from the Sun (A to B)
The planet moves more quickly when closest to the Sun (C to D)
Section 7.6

48. Kepler’s Third Law
The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.
T is the period, the time required for one revolution
T2 = K a3
For orbit around the Sun, K = KS = 2.97x10-19 s2/m3
K is independent of the mass of the planet
Section 7.6

49. Kepler’s Third Law, cont
Can be used to find the mass of the Sun or a planet
When the period is measured in Earth years and the semi-major axis is in AU, Kepler’s Third Law has a simpler form
T2 = a3
Section 7.6

50. Communications Satellite
A geosynchronous orbit
Remains above the same place on the earth
The period of the satellite will be 24 hr
r = h + RE
Still independent of the mass of the satellite
Section 7.6